4 Name Date 7.1 Notetaking with Vocabulary For use after Lesson 7.1 In your own words, write the meaning of each vocabulary term. monomial degree of a monomial polynomial binomial trinomial degree of a polynomial standard form leading coefficient closed Notes: 06 Algebra 1 Copyright Big Ideas Learning, LLC

5 7.1 Notetaking with Vocabulary (continued) Core Concepts Polynomials A polynomial is a monomial or a sum of monomials. Each monomial is called a term of the polynomial. A polynomial with two terms is a binomial. A polynomial with three terms is a trinomial. Binomial Trinomial 5x + x + 5x + The degree of a polynomial is the greatest degree of its terms. A polynomial in one variable is in standard form when the exponents of the terms decrease from left to right. When you write a polynomial in standard form, the coefficient of the first term is the leading coefficient. leading coefficient degree x 3 + x 5x + 1 constant term Notes: Extra Practice In Exercises 1 8, find the degree of the monomial. 1. 6s. w abc x y 6. 4r st 7. 10mn

12 Name Date 7.3 Special Products of Polynomials For use with Exploration 7.3 Essential Question What are the patterns in the special products a + b a b, a + b, and a b? ( )( ) ( ) ( ) 1 EXPLORATION: Finding a Sum and Difference Pattern Work with a partner. Write the product of two binomials modeled by each rectangular array of algebra tiles. a. ( x + )( x ) = b. ( x )( x ) = EXPLORATION: Finding the Square of a Binomial Pattern Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Draw the rectangular array of algebra tiles that models each product of two binomials. Write the product. a. ( x + ) = b. ( ) x 1 = 14 Algebra 1 Copyright Big Ideas Learning, LLC

18 Name Date 7.4 Solving Polynomial Equations in Factored Form (continued) 3 EXPLORATION: Special Properties of 0 and 1 Work with a partner. The numbers 0 and 1 have special properties that are shared by no other numbers. For each of the following, decide whether the property is true for 0, 1, both, or neither. Explain your reasoning. a. When you add to a number n, you get n. b. If the product of two numbers is, then at least one of the numbers is 0. c. The square of is equal to itself. d. When you multiply a number n by, you get n. e. When you multiply a number n by, you get 0. f. The opposite of is equal to itself. Communicate Your Answer 4. How can you solve a polynomial equation? 5. One of the properties in Exploration 3 is called the Zero-Product Property. It is one of the most important properties in all of algebra. Which property is it? Why do you think it is called the Zero-Product Property? Explain how it is used in algebra and why it so important. 0 Algebra 1 Copyright Big Ideas Learning, LLC

19 7.4 Notetaking with Vocabulary For use after Lesson 7.4 In your own words, write the meaning of each vocabulary term. factored form Zero-Product Property roots repeated roots Core Concepts Zero-Product Property Words If the product of two real numbers is 0, then at least one of the numbers is 0. Algebra If a and b are real numbers and ab = 0, then a = 0 or b = 0. Notes: 1

26 Name Date 7.5 Notetaking with Vocabulary (continued) In Exercises 13 18, solve the equation. 13. g 13g + 40 = k 5k + 6 = w 7w + 10 = x x = r 3r = 18. t 7t = The area of a right triangle is 16 square miles. One leg of the triangle is 4 miles longer than the other leg. Find the length of each leg. 0. You have two circular flower beds, as shown. The sum of the areas of the two flower beds is 136 π square feet. Find the radius of each bed. 8 Algebra 1 Copyright Big Ideas Learning, LLC

27 7.6 Factoring ax + bx + c For use with Exploration 7.6 Essential Question How can you use algebra tiles to factor the trinomial + + ax bx c into the product of two binomials? 1 EXPLORATION: Finding Binomial Factors Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to write each polynomial as the product of two binomials. Check your answer by multiplying. Sample x + 5x + Step 1 Arrange algebra tiles that Step model x + 5x + into a rectangular array. Use additional algebra tiles to model the dimensions of the rectangle. Step 3 Write the polynomial in factored form using the dimensions of the rectangle. width Area = x + 5x + = (x + )(x + 1) length a. 3x + 5x + = 9

31 7.6 Notetaking with Vocabulary (continued) 13. 1b + 5b x + x g 11g d d r 4r x 14x The length of a rectangular shaped park is ( 3x + 5) miles. The width is ( x + 8) The area of the park is 360 square miles. What are the dimensions of the park? miles. 0. The sum of two numbers is 8. The sum of the squares of the two numbers is 34. What are the two numbers? 33

32 Name Date 7.7 Factoring Special Products For use with Exploration 7.7 Essential Question How can you recognize and factor special products? 1 EXPLORATION: Factoring Special Products Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to write each polynomial as the product of two binomials. Check your answer by multiplying. State whether the product is a special product that you studied in Section 7.3. a. 4x 1 = b. 4x 4x + 1 = c. 4x 4x = d. 4x 6x + = 34 Algebra 1 Copyright Big Ideas Learning, LLC

33 7.7 Factoring Special Products (continued) EXPLORATION: Factoring Special Products Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to complete the rectangular arrays in three different ways, so that each way represents a different special product. Write each special product in standard form and in factored form. Communicate Your Answer 3. How can you recognize and factor special products? Describe a strategy for recognizing which polynomials can be factored as special products. 4. Use the strategy you described in Question 3 to factor each polynomial. a. 5x + 10x + 1 b. 5x 10x + 1 c. 5x 1 35

39 7.8 Notetaking with Vocabulary For use after Lesson 7.8 In your own words, write the meaning of each vocabulary term. factoring by grouping factored completely Core Concepts Factoring by Grouping To factor a polynomial with four terms, group the terms into pairs. Factor the GCF out of each pair of terms. Look for and factor out the common binomial factor. This process is called factoring by grouping. Notes: Guidelines for Factoring Polynomials Completely To factor a polynomial completely, you should try each of these steps. 1. Factor out the greatest common monomial factor. 3x + 6x = 3x( x + ). Look for a difference of two squares or a perfect square trinomial. x + 4 x + 4 = ( x + ) 3. Factor a trinomial of the form ax + bx + c into a product 3x 5x = 3x + 1 x of binomial factors. ( )( ) 4. Factor a polynomial with four terms by grouping. x 3 + x 4x 4 = ( x + 1)( x 4) Notes: 41

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